Sensitive dependence of Gibbs measures at low temperatures

TL;DR

This paper demonstrates that Gibbs measures for certain classical lattice systems can exhibit highly sensitive dependence on small perturbations at low temperatures, leading to significant changes in their behavior.

Contribution

The paper provides the first examples of sensitive dependence of Gibbs measures in classical lattice systems with smooth and Lipschitz continuous interactions at low temperatures.

Findings

Sensitive dependence observed in one-dimensional XY models.

Sensitive dependence shown for finite-state lattice systems.

Perturbations in smooth and Lipschitz topologies cause significant measure changes.

Abstract

The Gibbs measures of an interaction can behave chaotically as the temperature drops to zero. We observe that for some classical lattice systems there are interactions exhibiting a related phenomenon of sensitive dependence of Gibbs measures: An arbitrarily small perturbation of the interaction can produce significant changes in the low-temperature behavior of its Gibbs measures. For some one-dimensional XY models we exhibit sensitive dependence of Gibbs measures for a (nearest-neighbor) interaction given by a smooth function, and for perturbations that are small in the smooth category. We also exhibit sensitive dependence of Gibbs measures for an interaction on a classical lattice system with finite-state space. This interaction decreases exponentially as a function of the distance between sites; it is given by a Lipschitz continuous potential in the configuration space. The perturbations are small in the Lipschitz topology. As a by-product we solve some problems stated by Chazottes and Hochman.